An Analytical Discrete Ordinates Solution for the Two-Dimensional Transport Equation with Exact Representation of the Scattering Law

The mathematical complexities of solving the Boltzmann equation, even in the linear case, are well known. One of the most difficult challenges lies in treating the collision integral operator; in some applications, such as optical tomography modeling, the effects of scattering in biological tissues are a relevant factor to be considered. Recently, the Analytical Discrete Ordinates (ADO) formulation, combined with nodal techniques in two-dimensional geometries, was applied to the solution of problems in media with an arbitrary order of anisotropy. The formulation still provides explicit solutions in terms of the spatial variables for the average quantities generated by the nodal scheme (average angular flux or average radiation intensity).In this work, the ADO-Nodal method, as the two-dimensional formulation is commonly called, is extended to the solution of two-dimensional problems using the exact representation of the scattering law (referred to as the phase function in radiation transfer applications). We provide data comparing the use of the exact representation of the scattering law and the traditional Legendre polynomial expansions, and the analysis demonstrates computational gains with the exact representation. Numerical results are also presented for a problem with parameters used in optical tomography, indicating the need for representation in higher-order discrete ordinates. In addition, a diamond difference scheme was implemented to solve a test problem to compare with the proposed formulation. Furthermore, the eigenvalue problem, which is key to the ADO method, is derived in a general form, allowing for applications in other fields of particle transport.